The complement of a set is described as A’ = {x: x ∈ U and x ∉ A}

where,

A’ stands for the complement.

1. Complement Laws: The union of a set A and its complement A’ allows the universal set U of which, A and A’ are a subset.

A ∪ A’ = U

Also, the intersection of a set A and its complement A’ cause the empty set ∅.

A ∩ A’ = ∅

For Example: If U = {11, 12 , 13 , 14 , 15 } and A = {11 , 12 , 13 } then A’ = {14 , 15}. From this it can be seen that

A ∪ A’ = U = { 11 , 12 , 13 , 14 , 15}

Also, A ∩ A’ = ∅

2. Law of Double Complementation: According to the law, if we take the complement of the complemented set A’ then, we get the set A itself.

(A’)’ = A

In the previous example we can see that, if U = {11 , 12 , 13 , 14 , 15} and A = {11 , 12, 13} then A’ ={14 , 15}. Now if we consider the complement of set ‘A’ we get,

(A’)’ = {11 , 12 , 13} = A

This gives back the set A itself.

3. Law of empty set and universal set: According to this law the complement of the universal set gives us the empty set and vice-versa i.e.,

∅’ = U And U’ = ∅

This law is accessible.